Arithmetic Mean | Geometric Mean | Harmonic Mean

See all my videos at www.zstatistics.com/
0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question

[@martinplasse174]

If you can't explain something simply and clearly you don't truly understand it. This guy truly understands statistics. A lot people, including most teachers I would venture, have become proficient at statistics without a genuine understanding of what they are doing. Thx for this.

[@cannaroe1213]

I have worked at several top research institutes across Europe and the UK, and most aren't even proficient at statistics. Or rather, they've been brainwashed into thinking the arithmetic mean is somehow more important/special than the geometric mean or any other kind of statistic. It results in the publication of statistics everyone knows you should have but no one really knows what to do with it.

[@isha8229]

Accurate. never really understood the concept of GM or HM and this guy comes along! Could not be more interested in stats now. Thank you @zedstatistics

[@user-xm4ci6bp4b]

@@cannaroe1213 This is so because most statistics professors can calculate an arithmetic mean but have no clue what it actually means! :-) It should be called "level term", not arithmetic mean because it has nothing to do with the middle of anything.

[@WillOfHeaven]

@@cannaroe1213 This is the understanding that even I came onto on encountering his videos from a senior of mine who said that this would be beneficial for me

[@vuhuynh7629]

I'm two years too late to the party. But I'd like to put up my answer anyway. A superb video btw. Really appreciate it. I'll do it in 2 steps. Step 1: Find the average speed for stage 1. Since the two activities (the climb and the descent) are of the same distance but at different speeds, the average speed for stage 1 must be the harmonic mean of two speeds. Average speed (stage 1) = 2 / (1/10 + 1/50) = 16.67 km/h. Step 2: Find the average speed between stage 1 and stage 2. Since stage 1 and stage 2 involve the same amount of time (both are 30 minutes long), the average speed would be the arithmetic mean. Average speed (stage 1 & 2) = (16.67 + 40) / 2 = 28.33 km/h. I'm sure others already came up with the perfect answer long ago. Still, I wanted to try myself. Thanks a ton for the examples. They really put all the different types of means in perspective.

[@MohamedEl-Gohary-gd5nu]

well, I'm 2 years behind you and came to check my answer and it is similar to yours. you are never too late!

[@Raudadrosin]

I really are the best statistic teacher EVER! How deep you go in explaining everything is finally making me understand... I´ve been so lost because I didn´t have the foundation for the understanding! Both the lectures + how you talk and describe = perfect in every way! I really hope that I will finally pass my statistics esam this Saturday!

[@endubi]

The best statistics teacher ever! You break down complex topics into simple terms that can easily be understood.

[@tsvianshell8189]

You don't have to use harmonic mean, a weighted mean is also possible: On stage one, since uphill distance equals downhill distance (same hill) - the times spent on every part can be found using the equation: 10X=50(30-X) which gives X=25. The next stage is just a weighted mean: (10*25+50*5+40*30)/60=28.33

[@hanswurst4728]

Very nice, thank you.

[@divine6104]

But we can't assume the distance uphill is the same as the distance downhill, can we? The slopes of the hill may vary. I'm aware I'm one year late tho lol

[@virgiliomurilloochoa2884]

totally wrong

[@jonathanyun7817]

​@@virgiliomurilloochoa2884totally unhelpful

[@subhrajitchakraborty8170]

Thank you for this viewpoint, it was helpful 👍🏾

[@rohit-rv8cr]

We will find harmonic mean of two activities of stage one and then we will find arithmetic mean for both the stages. so, harmonic mean of stage one = 2/(1/10+1/50)= 16.67. Now the arithmetic mean for both the stages = (16.67+40)/2=28.33 His average speed is 28.33 Km/h. btw, great video i was confused why we use geo mean. You were excellent.

[@naveentogu8526]

Thanks mate.... I was wondering, if I was wrong.

[@delcydadelia728]

Yess

Can we combine 02 stages in one because the stage 1 & 2 have the same time 30 minutes without rest or something else. Hence we only find the harmonic mean of 3 activities = 3/(1/10+1/50+1/40) = 20.68 ?

[@XBox360Quest]

@ That's what I did! to me, it makes no sense to calculate the arithmetic mean

[@re-Rina]

Is this assuming that he descends the same distance he's climbed?

[@cannaroe1213]

Such a great video! Square-rooting finally clicked for me, because I always think of it as returning a single number, but it's actually returning 2 numbers, or 3 for cube-root, but because all the numbers/dimensions are the same we just refer to the product as a single number. Applying that to ROI was great! Maybe at 14:04 it would help some people to know 1 Hertz is 1 thing per second, so that's where the ratio is coming from, the fact that you are talking about 2 dimensional frequencies, oscillations in time, but referring to it as a single number. Harmonic was explained so well i wish i could subscribe twice!

[@seekingalpha_]

i cant thank you enough for this series. Your explanation is literally so on point it couldn't be anymore perfect than this!

[@arashchitgar7445]

first stage: we have fixed distance, so harmonic mean for km/h: (1/10)+(1/20) = 16.66 1stage + 2nd stage: we have a fixed time, so arithmetic mean for km/h: (16.66 + 40) / 2 = 28.33

[@007Anukul]

yup, that's the right answer, mine is same.

[@santiagoleon7475]

except, the descent speed is 50kmh not 20, so the answer is 26.67

[@stacyliddell5038]

@@santiagoleon7475 not true. the answer is 28.33. Check your work again.

[@JonnyBurkholder]

@@santiagoleon7475 I think it was just a typo on his part. If you do the math out with 1/50 instead of 1/20, you get 28.33

[@shykatislam5581]

@@JonnyBurkholder you are right..............i was confused ......where he got 1/20..........

[@LayneSadler]

could be cool for measuring ml model improvement over the course of either epochs or time during training

[@supersql8406]

Brilliant! Straight to the point of the content and backup with beautiful explanation and example!!

[@chisaomusician7752]

Appreciating the detailed definitions thanks.

[@economicskibaat8013]

Thank you for the brilliant explanation

[@cw9249]

thanks. i didnt think i'd learn something new, but i did. and i loved the question in the end. the key is recognizing that the first part of the problem didnt specify how much time were spent cycling the two different speeds, but it was given to you how much distance were spent cycling the two different speeds, so use the harmonic mean!

[@Raymonddoc1]

A gift that is exceptional and truly educational. Thank you.

[@dushimumuremyibertrand3263]

I was oblivious of the difference between these three terms, but after watching this, I clearly understand each one of them. Thank you so much!

[@radhakrishntextile8020]

Thankyou for bringing conceptual clarity

[@AJ-fo3hp]

Thank You for your explanation Simple example, simple sentences, crystal clear. There is one more is there Root Mean Square Value(RMS)

[@denki2558]

Stage 1 - HM = 2/(1/10+1/50) = 50/3 ≈ 16.66 Overall Average - AM = (HM + 40) / 2 = 20 + 25/3 = 85/3 ≈ 28.33 units are in kmph Edit for checking: let x = time taken to ride uphill in hours since downhill took 5 times faster than uphill, then 0.5-x=x/5 -> x=25/60 (25mins) Total Distance = 10x + 50(0.5-x) + 40*0.5 = 25/6 + 25/6 + 20 = 85/3 ≈ 28.33 Total Time = 0.5 + 0.5 = 1 (1 hour) Average Speed ≈ 28.33 / 1 ≈ 28.33

[@echelonrank3927]

hmmm, ok, but i dont see how knowing the average speed is of any use. the same way i dont see how knowing the average shoe size helps sell more shoes. they are there 1 day and out of stock 99 days. useless.

[@tatiana1310]

Brilliant! Thank you so much. I felt dumb because I was unable to solve the final question but I understood the solution in the comments.

[@christopherlinder5045]

Great video. Short enough for an intro, but long enough for some very helpful details. I came across your video after recently having seen a question along the lines of "Ben takes 2 hours and 12 minutes to dig a hole, Bob beats him and digs the same hole in 1 hour and 50 minutes, how long would they have taken to dig the hole together in joint effort?" which involves basically half of the harmonic mean, in this case, and that got me interested. It also reminded me of the fact that runners often use "pace", which is min/km, rather than, say speed in km/h. So a pace of 6 would be 10 km/h, and a pace of 8 would be 7.5 km/h. Essentially, since one is having to take the reciprocal in order to add the speeds before one flips it back into pace with another reciprocal, these types of questions of "resultative times in combined effort" are related to the harmonic mean. Oddly though, I have to remark that with musical notes and their frequencies, at least with well-tempered tuning like the piano, where let us say A = 440 Hz and the frequencies of each semitone higher is the precursor frequency multiplied by 12th root of 2 which is around 1.0595 or 5.95% higher, this entails that finding the frequency of the notes in between two notes involves the geometric mean. So whereas the arithmetic mean of the 440 Hz A and the 880 Hz A would be an E of 660, which is "too high" because it is a natural fifth away from the low A and only a natural fourth away from the high A, nicely shown by it being 7 semitones away from the low note and 5 semitones away from the high note, if you want the D# or E flat that's 6 semitones away (a tritone) from both, you can take the geometric mean of sqrt( 440 x 880 ) and you will get exactly 440 times sqrt(2), which is around 622.25 Hz. (Of course, with waves and music being what they are, A-E-A (Strauss's Zarathustra) will always sound great, and like "the middle" because it's consonant and A - D# - A (Holst's Planets, Mars) will always sound scary because its a tritone and dissonant, but such is the odd design of the keyboard, and tempered tuning is mathematically always a "compromise" of sorts.

[@petercourt]

Beautiful explanation, thank you!

[@jbhonest]

Perfect. Thank you.

[@TheScawer]

First time I really understand harmonic mean, thank you! Your way of inverting it (twice) really helps with the understanding.

[@petercole2592]

With the harmonic mean, I am a financial stats students at uni and just getting my head around functions. I understand mean standard deviation, variance deducting mean from values and square roots, but I cant understand why (1/3 + 1/2)/2 becomes 2/(5/6) then 12/5 I can calculate it as 2.4 on a calculator, but I don't understand those fractions. I am only 3 weeks into Statistics, so algebra functions are just being learned for me

[@deogettic]

Great challenge activity. Great twist. This is very interesting

[@Salah.alkhalifa]

Best video on this subject! Greetings from Eastern Arabia

[@pfever]

Challenge question: What is the average speed over the whole track? Ans. 28.33 km/h Stage 1.- 30 mins - up-hill 10km/h, down-hill 40km/h, which means, equal distance traveled in a different time, therefore use harmonic mean. Stage 2.- 30 mins - 40km/h, equal time as stage 1, so use the arithmetic mean between them. average of stage 1: ( (1/10 + 1/50) / 2 )^-1 = 16.66 km/h average of stage 1 and stage 2: (16.66 + 40) / 2 = 28.33 km/h

[@hhhhhhahah]

Harmonic Mean for Stage One = 16.6km/h, Stage two mean is fix to 40km/h. I then computed the arithmetic mean: *28.3 km/h* is this correct?

[@SiddharthPrabhu1983]

I got the same answer. Is it right?

[@saqibshams5143]

I got the same answer 28.3km/h

[@winstonsmith3373]

It’s not weighted yet, as you’re actually going at an average speed of 16.67 km/h for a total of 36 minutes (when the down hill ride is included) in contrast to the 30 minutes ride at 40 km/h. Hence, if you add the proper weight (by using factor [36/30 =] 1.2 for the first stage), you can afterwards take the arithmetic mean of both values by deviding the sum by 2.2 (instead of just two). That get’s you a result of [(stage1*1.2+stage2)/2.2 =] 27.27… km/h. ;)

[@TheGeneralBullet]

@@winstonsmith3373 are you sure the 30mins don't include the downhill? It seems like it's total

[@ahmadnomanalnoor5986]

yes to all those who got 28.3

[@kundanshekhawat6929]

Enjoyed the way you teach. Thnaks

[@007Anukul]

In the first part, the mean is across 2 equal distances (n = 2). So, ((1/10)(1/50)/2)^-1 = 16.67 The mean along the first and second part is across equal time, so it is not GM but AM so (16.67+40)/2 = 28.33

[@jt101010]

Hi, great video, thanks so much. I'm curious, has anyone observed the connection between these three means and special relativity before? Each mean when applied to light paths between points derives a fundamental kinematic concept (proper length/time, time, length), but I haven't seen this explained anywhere! This video helped clarify these means for me, so much thanks!

[@rrrprogram8667]

Subscribed... Keep up the good work mann

[@bethebestfucktherest]

Great video. Thanks.

[@melodykamil]

Great Video. Thank you

[@elanjelian]

It appears harmonic mean only works if the numerator in the rate or ratio is the same across the values. In the above example, the cycling speeds up and down the hill are given as 10 km/h and 50 km/h respectively. We could determine the average speed using the harmonic mean formula only because the distance covered both ways is the same. Should the distance be different, and the speeds be the same as provided above, we may not use the harmonic mean formula to determine the average speed, it appears.

[@keerthymariamnebu6976]

Wow, very clearly explained!

[@vickyanga]

Exceptional explanation for a lifetime

[@shenay9377]

That was exactly what i needed, thank you.

[@srihariharana7143]

Thanks a lot I found it very useful and explanation with real time example are quite good and impressive

[@taherhussain2]

Really genius in the clear explanation

[@samarthsai9530]

Really good ...amazing.

[@thomasbates9189]

What an amazing explanation of geometric mean!

[@MrIkmls]

any video on geometric growth?

[@jayasreekrishna646]

Thanks for this!

[@channel_panel193]

Seems like we know to use a geometric mean, when the effects of each step effect the next step. Like the Tesla example. In these cases, summation isn't the right tool.

[@CHARUSREE156]

Thank you so much.i Cleared my doubts.

[@nikitagiri1111]

This is amazing!! Thank you!!! I have subscribed!

[@jeevacation]

Thanks so much for this man

[@alzadid4837]

I sent this video to a friend of mine and captioned: The best introductory video on stat means I've seen.

[@adwithiyapachisia3061]

Stage 1 Time taken- 30mins Hill climb- 10km/hr Descent- 40 km/ hr So average speed in stage 1= Harmonic mean of both the different speeds of stage 1 2/6/50 = 2×50/6 = 16.666 Stage 2 Time taken- 30 mins Speed- 40km/hr Now which means Stage 1 average speed is 16.666 km/hr And stage 2 average speed is 40 km/hr Average speed of the both stages combines = Arithmetic mean of stage 1 speed and stage 2 speed 40+16.6666= 56.666 and then Divided by 2 bcoz ( see formula ) =56.666/2 = 28.333( final ans )

[@rrrprogram8667]

Briilianttt.... I never remembered the use case of geometric and harmonic mean... But now i dooo... Great stufff

[@pisquie]

Wonderful video - thanks! Never heard of harmonic mean before... I got the result 36.7 km/h as average speed.

[@thebreedersassistant2785]

I just can't thank you enough! Either you are the best at explaining this or I woke up smarter today. I think it is the former.

[@tanvedzmakkar231]

How much your tesla stock worth now?

[@gauravbansal148]

Challenge question: its been answered by many people but i would like to add another way. It is also possible to get the distance of the hill and time taken both uphill and downhill easily by solving distance/time equals speed equation. The answer comes out to hill being 4.17km each way and taking 25min uphill and 5min downhill. We still get the same final speed of 28.33.

[@redcat7467]

Cool explanation of a geometric mean meaning.

[@mohammedkabir4030]

This Question is very interesting and clever, because I believe that the entire calculation method to get answer involves usage of all 3 types of mean we learned here....here we go, Step 1....We need to do Harmonic mean between ascent & descent values of stage one ride (because the distance is same, rate may differ) Step 2....We need to Geometric mean between (Step 1 answer value) & Stage 2 value....because, the distance and time are different. Step 3....Simple Arithmetic Mean between values of Step 1 & Step 2 I really not bother to know correct answer value, rather to know, how to understand this question and method of calculating it practically.....Any other thoughts ?? Justin, pls correct me if my understanding is not correct.

[@ahmadnomanalnoor5986]

close, but not quite.you don't actually need to use the geometric mean here.

[@ThugStrike]

Thank you

[@varshavarsha5790]

Nice work

[@divine6104]

Correct me if I'm wrong, but isnt the challenge problem flawed? In stage one, we cannot assume the distance uphill equals the distance downhill- we know only that the sum of the time taken to go uphill is 0.5hrs ie: t1+t2=0.5, and that 10t1 + 50(0.5-t1)= total distance travelled in stage one denoted by D1. There's no way to link any component of D1 to other bits of information given. Therefore, there are 2 degree of freedom (ie: (t1 or t2) and a component of D1): there's no unique solution.

[@TheKhashayarArt]

Your accent makes me feel I'm a student of Mathematical Institute, University of Oxford :) love it!

[@user-uq2ms3kp3t]

first of all stage 1 we have to find arithmetic mean of both 10 km/h and 50 km/h so the value of them well be 29.7 km/h then we can make a triangle with side 1 =10 km/h side 2 =50 km/h side 3 =29.7 km/h then we find the Geometric mean of stage 1 which is 24.264 km/h so we can make a Harmonic mean of both stages which is 30.441 km/h

[@gloriouz89]

Thank you for your video! Question regarding Geometric Mean. Why use Geometric Mean over just taking the average of 10%,20%, and 30%? Is there a benefit to using Geometric mean?

[@zedstatistics]

If your investment earned 5% this year, 8% last year and 11% the year before, what is the average yearly rate of interest across the entire life of your investment? :)

[@evan.5967]

@@zedstatistics what????

[@evan.5967]

@@zedstatistics isn't that effective interest

[@parmindr001]

@@zedstatistics 7.9%

[@anjishnu8643]

Arithmetic mean doesn't consider the effect of compounding (of the investment over time). But Geometric mean does. Say you made a profit of 10% in year 1, loss of 20% (or, -20%) in year 2 and a 10% profit in year 3 on a $1000 you had invested at the very beginning. Looking at how the investment changed over the years, Investment at the beginning of year 1 = $1000 At the end of year 1 = 1000 + 1000 * 0.10 = 1000 * (1 + 0.1) = 1000 * 1.1 = $1100 At the end of year 2 = 1100 + 1100 * (-0.2) = 1100 * (1 - 0.2) = 1100 * 0.8 = $880 At the end of year 3 = 880 + 880 * (0.1) = 880 * 1.1 = $968 The arithmetic mean for the rate of return is = [0.1 + (-0.2) + 0.1] / 3 = 0 The geometric mean for the same is = cubeRoot(1.1 * 0.8 * 1.1) = 0.989 Had the average rate of return actually been zero, the investment at the end of year 3 would have been $1000 itself (same as your initial investment), which is not the case. The geometric rate of return being 0.989 signifies that there was a (1 - 0.989) * 100 = 1.1% loss on average which caused the $1000 to end up at $968. Now from what I understand, since arithmetic mean is always greater than or equal to the geometric mean, it could be used by businesses to deceive unknowing investors showing them a higher rate of return on their sales.

[@asjednazir6582]

thank u so much

[@christopherl-b1892]

What does n mean in the harmonic mean? How many times you went swimming?

[@sergiomarinhodasilva4781]

Distance crossed: 25/60 min * 10 km + 5/60 min * 50 km + 30/60 min * 40 km = 28.33 km. Since the travel was 1h, it was 28.33 km/h. Is this right? How would I resolve it using harmonic mean? or any other mean?

[@brunovillagomezcaro388]

i have a question , if we are sampling do we still use n or n-1

[@splinter471]

Probably obvious to most or all, for tossing it out there anyway. If using the GEOMEAN formula in excel, DON'T consider year 0 in the range.

[@prathamsinghal5261]

28.34 km/hr

[@PHYSICSCHEMTutor]

How do we find the missing terms in a set given the 3 means

[@asifhasan8752]

30 min @10km/hr. That’s 5 km in 30 min. 5km back @50km/hr. 5 km in 6 min. 30 min @40km/hr. 20 km in 30 min. Total distance 30 km Total time 66 min Avg speed (30/66)*60=27.272727.... Km/hr

[@MeetPatel-ps1hh]

stage 1 is 30mins total, yours is 36 mins

[@jvan99ful]

How to calculate the adjusted mean?

[@anyaozioma9583]

Thanks 🙃

[@_aditya_9416]

I just had a question that can any one find length of the lap with given values of Harmonic mean??

[@aedaldaniel]

How did I know that I'd have to do harmonic mean in stage 1 and then the arithmetic mean of state 1 and stage 2 get the average speed?

[@yvonnemwangi8004]

Saved my life!!!!

[@MrIgottheway]

First stage: harmonic mean to 10km/h and 50km/h = (sum(1/10 + 1/50) /2)^-1 = 16.66km/h Second stage: harmonic mean to firststage and second = (sum(1/16.66 + 1/40) /2)^-1 = **23.52 km/h**

[@Makkhan-xk3nx]

Subscribed

[@tvvt005]

8:51 hi ims not able to understand the logic behind removing that one,,I’m also confused about how we bring in the one in the first place..

[@bharathinila656]

Hmm. I have a dbt??? Is weighted arihmetic mean and Expected value the same!!

[@ahmadnomanalnoor5986]

well im not sure about this but the expected value is the weighted arithmetic mean in the context of distributions and random variables. for example: the expected value for 10 rolls of a fair die is 35, this will always be the case for 10 rolls. suppose after rolling the die 10 ties, you got the values (1,4,2,1,5,3,6,2,3,5) the weighted arithmetic mean for this sample is 3.2, this will be different for each combination of values.

[@bharathinila656]

Thank you brother.

[@hassanabdi8897]

here is my try you have speed for both the climbing and decent of the hill.....10km/h & 50km/h you also have the total time taken on the hill....30min you have the speed on the flat ground 40km/h and the total time on the flat ground 30min to find the average speed you find the total distance covered over the span of the total time taken which is 1hr don't assume that the distance of climbing the hill is equal to the decent distance so, find the geometric mean of the speeds on the hill which is approximately...22.36km/h now the total distance of the hill is....11.18km the distance of the flat ground is.....(D*T).....(40km/h * 0.5hrs)....20km average speed is (11.18+20)/1= 31.18km/h

[@DanielL143]

Excellent, again an above Average video but the challenge was a bit Mean (28.3 km/hr).

[@user-xm4ci6bp4b]

@3:16 The mean in a normal density function should really be called a *median*, not an arithmetic mean because in the distribution itself, this value corresponds to a frequency that is most likely, but definitely not the mean. To convince yourself of this fact, find the average value (using calculus) of the frequencies between one standard deviation, that is, -1 to 1 (if normalised). You will observe that there are two arithmetic means and they are not located at the median. The phrase arithmetic mean is not a very well thought out concept. A better expression would be "level term" because that's what you do when you calculate and arithmetic mean - you redistribute the values so that they are all equal. This is why it makes no sense to find a level term for the height of a sample of people: what? are you going to chop them up and then glue them all back together again in such a way that they are all of the same height? We can know beforehand whether it makes sense to calculate a level term or not. Simply this: A level term only makes sense if redistribution makes sense. It should be called "level term", not arithmetic mean because it has nothing to do with the *middle of anything*

[@yulinliu850]

Much appreciated!

[@tompeter8890]

28.33

[@rahulmascarenhas1370]

28.34

[@user-bd1bl8ue9g]

FOR CHALLENGE PROBLEM: why is it wrong to calculate it using (total distance travelled)/(total time taken)?

[@hailothanks523]

Stage one mean ===> (2/(1/10 + 1/50) = 16.67 km/h Stage1 + Stage2 Mean ===> 2/(1/16.67 + 1/40) = 23,53 km/h

[@naixin-lg6tc]

(16.67+40)/2=28.33

[@parmindr001]

No need for harmonic mean in 2nd stage as time is same

[@onurkula5756]

Why cannot we use harmonic mean in case of a dataset which contains negative values

[@kirangandi9923]

50

[@tutortravel1731]

28.33 km/h

[@asanthaharshana1885]

28.33km/h

[@nimadashtban8258]

28.332

[@jennifershao4608]

28.3km/h.

[@cutetiny6144]

25 kmph is that right.

[@yuliefelice5643]

❤

[@tonib4701]

I'm trying to calculate harmonic mean of 6/((1/0.5)+(2/-1.2)+(3/-0.75)) with no success because of the negative numbers (-1.2 & -0.75)...any help would be much appreciated

[@ahmadnomanalnoor5986]

this doesn't look anything like the harmonic mean. can you tell the figures you are trying to calculate the harmonic mean of?

[@VS-yc1lu]

Wouldn’t you take the absolute value of both the negative numbers? If you think about it as distance over time then you need to take the absolute value since distance can never be negative. So even if you’re trying to measure something else it should the absolute value of the change over the time the change occurred

[@scratchmyback246]

36.67 maybe ?!?

[@rishitshah4087]

Sir I have one doubt sir in harmonic mean Harmonic Mean Query Example: we travel 60 km at 60 km/h, than another 20 km at 20 km/h, what is our average speed? Harmonic mean = 2/( 1/60 + 1/20 ) = 30 km/h Check: the 60 km at 60 km/h takes 60 minutes, the 20 km at 20 km/h takes 60 minutes, so the total 80 km takes 120 minutes, which is 40 km per hour which one is rigth ?

[@vuhuynh7629]

Remember this, mate. You compute the harmonic mean when you travel the same distance at two different speeds. You cannot work out the harmonic mean in your example because the distances are different (60km at 60km/h; then 20km at 20km/h). The harmonic mean of 30km/h, as you've computed, would apply to the situation where you travel 60km at 60km/h; then another 60km at 20km/h.